p-adic Cherednik algebras on rigid analytic spaces

Let $X$ be a smooth rigid space with an action of a finite group $G$ satisfying that $X\mathrm{/}G$ is represented by a rigid space. We construct sheaves of $p$-adic Cherednik algebras on the small étale site of the quotient $X\mathrm{/}G$, and study some of their properties. The sheaves of $p$-adic Cherednik algebras are sheaves of Fréchet-Stein $K$-algebras on $X\mathrm{/}G$, which can be regarded as $p$-adic analytic versions of the sheaves of Cherednik algebras associated to the action of a finite group on a smooth algebraic variety defined by P. Etingof.